Multistage sampling example

What it shows

Multistage sampling takes samples in stages, with smaller sampling units at each stage. The diagram is a simplified, scaled-down version of the Australian Bureau of Statistics household survey design: 2 of 4 collection districts are drawn at random, then 2 of the 5 blocks in each selected district, then 3 of the 6 dwellings in each selected block.

The sample is 2 × 2 × 3 = 12 dwellings, and every dwelling has the same chance of selection, 2/4 × 2/5 × 3/6 = 1/10.

How to read it

1. 1

   Darker panels are the districts selected at stage 1.

2. 2

   Dark outlines mark the blocks selected at stage 2, inside the selected districts only.

3. 3

   Filled circles are the dwellings selected at stage 3. Blocks are numbered only inside selected districts and dwellings only inside selected blocks, because each list is needed only there.

The numbers behind it

• Population of 120 dwellings: 4 collection districts, 5 blocks per district, 6 dwellings per block.
• Stage 1: districts 2 and 4. Stage 2: blocks 4 and 5 in district 2; blocks 1 and 3 in district 4. Stage 3: 3 of 6 dwellings in each selected block.
• Sample 2 × 2 × 3 = 12 dwellings; inclusion probability 2/4 × 2/5 × 3/6 = 1/10.
• Every stage was drawn with a seeded random number generator (mulberry32, seed 20261421), so the draws can be repeated.

Ways to use it in teaching

• Have students count how many dwellings had to be listed (24) against the 120 a simple random sample would need.

• Ask what changes if stage 3 takes all 6 dwellings in each selected block: the design becomes two-stage cluster sampling (districts, then whole blocks), and the sample doubles to 24.

• Change the numbers at one stage (say 3 of 5 blocks) and have students work out the new sample size and inclusion probability.

For when each method suits a study, and how to justify the choice in a methodology chapter, see our guide to sampling methods.

Sources

• Ahmed, S. (2009). Cluster sampling [Lecture 5, Methods in sample surveys, 140.640]. Johns Hopkins Bloomberg School of Public Health OpenCourseWare.
• Australian Bureau of Statistics. (2006). Labour statistics: Concepts, sources and methods (Cat. No. 6102.0.55.001). https://www.abs.gov.au/ausstats/abs@.nsf/mf/6102.0.55.001
• Multistage sampling. (n.d.). In Wikipedia. Retrieved October 19, 2026, from https://en.wikipedia.org/wiki/Multistage_sampling